Problem: Simplify the following expression and state the condition under which the simplification is valid. $r = \dfrac{-10n^2 + 100n - 210}{-2n^2 + 24n - 54}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ r = \dfrac {-10(n^2 - 10n + 21)} {-2(n^2 - 12n + 27)} $ $ r = \dfrac{10}{2} \cdot \dfrac{n^2 - 10n + 21}{n^2 - 12n + 27} $ Simplify: $ r = 5 \cdot \dfrac{n^2 - 10n + 21}{n^2 - 12n + 27}$ Next factor the numerator and denominator. $ r = 5 \cdot \dfrac{(n - 3)(n - 7)}{(n - 3)(n - 9)}$ Assuming $n \neq 3$ , we can cancel the $n - 3$ $ r = 5 \cdot \dfrac{n - 7}{n - 9}$ Therefore: $ r = \dfrac{ 5(n - 7)}{ n - 9 }$, $n \neq 3$